# Courses of Study : Mathematics

Modeling
Mathematical modeling and statistical problem-solving are extensive, cyclical processes that can be used to answer significant real-world problems.
 Mathematics (2019) Grade(s): 9 - 12 Mathematical Modeling All Resources: 0
1. Use the full Mathematical Modeling Cycle or Statistical Problem-Solving Cycle to answer a real-world problem of particular student interest, incorporating standards from across the course.

Examples: Use a mathematical model to design a three-dimensional structure and determine whether particular design constraints are met; to decide under what conditions the purchase of an electric vehicle will save money; to predict the extent to which the level of the ocean will rise due to the melting polar ice caps; or to interpret the claims of a statistical study regarding the economy.
Unpacked Content Evidence Of Student Attainment:
Students:
• Identify the problem to solve when given a real world situation.
• Identify relevant and non-relevant information in the problem.
• Apply appropriate mathematical procedures to solve the problem.
• Can identify restrictions or limits of their model.
• Can assess their own work and answer(s) to insure it is appropriate.
• Realize that they may have to refine or extend their model.
• Report their findings to others.
• Can apply their work to similar situations.
Teacher Vocabulary:
• Mathematical Modeling Cycle
• Statistical Problem Solving Cycle
• Mathematical Model
Knowledge:
Students know:
• how to approach and solve real-world problems using mathematical models and the mathematical modeling cycle.
Skills:
Students are able to:
• Read a real-world problem and distinguish between relevant and non-relevant information.
• Create and apply a mathematical model to solve a real-world problem.
• Analyze their work and their findings.
• note restrictions or limits that arise when using a mathematical model.
• Share their findings with others.
• Apply their work to similar situations.
Understanding:
Students understand that:
• real-world problems can be solved using a mathematical model.
• There is a defined process to follow to create a mathematical model for problem solving.
• Mathematical models sometimes have limitations or need to be restricted in order to produce valid answers.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
MMOD.1.1: Define the mathematical model and the statistical problem-solving cycle.
MMOD.1.2: Use the mathematical model or the statistical problem-solving cycle to solve a real-world problem.
MMOD.1.3: Determine which model to use, mathematical modeling or statistical problem solving, in a real-world problem.

Prior Knowledge Skills:
• Recall estimation strategies.
• Analyze the given word problem to set up a mathematical problem.
• Recall problem solving methods.
Financial Planning and Management
Mathematical models involving growth and decay are useful in solving real-world problems involving borrowing and investing; spreadsheets are a frequently-used and powerful tool to assist with modeling financial situations.
 Mathematics (2019) Grade(s): 9 - 12 Mathematical Modeling All Resources: 0
2. Use elements of the Mathematical Modeling Cycle to solve real-world problems involving finances.

Unpacked Content Evidence Of Student Attainment:
Students:
• Identify the problem to solve when given a real world financial situation.
• Identify relevant and non-relevant information in the problem.
• Apply appropriate mathematical procedures to solve the problem.
• Can identify restrictions or limits of their model.
• Can assess their own work and answer(s) to insure it is appropriate.
• Realize that they may have to refine or extend their model.
• Report their findings to others.
• Can apply their work to similar situations.
Teacher Vocabulary:
• Mathematical Modeling Cycle
• Mathematical Model
Knowledge:
Students know:
• how to approach and solve real-world financial problems using mathematical models.
Skills:
Students are able to:
• Read a real world financial problem and distinguish between relevant and non-relevant information.
• Create and apply a mathematical model to solve a real-world financial problem.
• Analyze their work and their findings.
• note restrictions or limits that arise when using a mathematical model.
• Share their findings with others.
• Apply their work to similar situations.
Understanding:
Students understand that:
• Real-world financial problems can be solved using a mathematical model.
• There is a defined process to follow to create a mathematical model for financial problem solving.
• Mathematical models sometimes have limitations or need to be restricted in order to produce valid answers.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
MMOD.2.1: Analyze a personal budget.
MMOD.2.2: Design a monthly budget, including investments, savings, borrowing and credit.
MMOD.2.3: Differentiate the various modes of payment options (cash, check, money order, debit cards, credit cards).
MMOD.2.4: Determine and prioritize personal needs and wants according to current or expected income (housing, food, clothing, transportation, wellness needs, healthcare, utilities, insurance, benefits).

Prior Knowledge Skills:
• Experience with checking and savings accounts.
• Real-world examples of credit cards
• Determine personal needs and contrast with wants.
• Analyze data from tables.
• Summarize categorical data for two categories in two-way frequency tables.
• Recognize possible associations and trends in the data.
• Create a scatter plot and line of best fit using data from a spreadsheet.
• Organize numerical data in a spreadsheet.
• Create graphical representations from classroom
• generated data to model consumer costs.
• Create graphical representations from classroom
• generated data to predict future outcomes.
• Create graphical representations from equations to model consumer costs.
• Create graphical representations from equations to predict future outcomes.
• Create graphical representations from tables to model consumer costs.
• Create graphical representations from tables to predict future outcomes.
 Mathematics (2019) Grade(s): 9 - 12 Mathematical Modeling All Resources: 0
3. Organize and display financial information using arithmetic sequences to represent simple interest and straight-line depreciation.

Unpacked Content Evidence Of Student Attainment:
Students:
• Given information about an investment or loan involving simple interest or Straight-line depreciation, can identify the initial value and the periodic rate of change.
• Can write the arithmetic sequence that will generate data for the problem.
• Can display the data in an organized manner and use it to answer questions about trends and results.
Teacher Vocabulary:
• Simple Interest
• Straight Line Depreciation
• Arithmetic Sequence
Knowledge:
Students know:
• how to select information from a real-world financial problem, such as the initial amount of the investment and its periodic rate of change, and use it to model simple interest and straight line depreciation.
Skills:
Students are able to:
• Identify the first term and the common difference in an arithmetic sequence.
• Recognize that the first term of an arithmetic sequence is the initial value of a loan or investment.
• Recognize that the common difference in an arithmetic sequence is the rate of change in the loan or investment.
• Use an arithmetic sequence to model simple interest or straight-line depreciation.
• Display data found by using an arithmetic sequence to model simple interest or straight-line depreciation.
Understanding:
Students understand that:
• The initial amount of an investment and its periodic rate of change correlate to the first term and the common difference in an arithmetic sequence.
• Arithmetic sequences can be used to model simple interest and straight-line depreciation.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
MMOD.3.1: Define arithmetic sequences, simple interest, and straight-line depreciation.
MMOD.3.2: Analyze the long-term costs of borrowing money.
MMOD.3.3: Calculate straight-line depreciation.
MMOD.3.4: Calculate simple interest.
MMOD.3.5: Identify the formula to compute straight-line depreciation.
MMOD.3.6: Identify the formula to compute simple interest.

Prior Knowledge Skills:
• Define interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, and percent error.
• Apply definitions to context in real world problems.
 Mathematics (2019) Grade(s): 9 - 12 Mathematical Modeling All Resources: 0
4. Organize and display financial information using geometric sequences to represent compound interest and proportional depreciation, including periodic (yearly, monthly, weekly) and continuous compounding.

a. Explain the relationship between annual percentage yield (APY) and annual percentage rate (APR) as values for r in the formulas A=P(1+r)t and A=Pert.
Unpacked Content Evidence Of Student Attainment:
Students:
• given information about an investment or loan involving compound interest or proportional depreciation, can identify the initial value and periodic rate of change.
• Can apply a geometric sequence to generate data for the problem.
• Can display the data in an organized manner and answer questions about the trends and results found.
• Can explain the relationship between APR (annual percentage rate) and APY (annual percentage yield) using compound interest formulas.
Teacher Vocabulary:
• Compound Interest
• Geometric Sequence
• Proportional Depreciation
• Periodic
• Annual Percentage Rate
• Annual Percentage Yield
Knowledge:
Students know:
• how to select information from a real-world financial problem, such as the initial amount of the investment and its periodic rate of change, and use it along with a geometric sequence to model compound interest and proportional depreciation.
Skills:
Students are able to:
• Identify the first term and common ratio in a geometric sequence.
• Recognize that the first term of a geometric sequence is the initial value of the loan or investment.
• Recognize that the common ratio is either (1+rate of growth) or (1-Rate of decay).
• Use a geometric sequence to model compound interest or proportional depreciation.
• Display data found using a geometric sequence to model compound interest or proportional depreciation. Relate APR to APY using compound interest formulas.
Understanding:
Students understand that:
• the initial amount of an investment or a loan and its periodic rate of change correlate to the first term and the common difference in a geometric sequence.
• Geometric sequences can be used to model compound interest and proportional depreciation.
• The annual percentage rate is the yearly rate of interest while the annual percentage yield is the rate you actually pay when compound interest is included.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
MMOD.4.1: Define geometric sequence, compound interest, proportional depreciation, frequent compounding, continuous compounding, annual percentage yield and annual percentage rate.
MMOD.4.2: Calculate proportional depreciation.
MMOD.4.3: Identify the formula for proportional depreciation.
MMOD.4.4: Calculate compound interest.
MMOD.4.5: Calculate simple interest.
MMOD.4.6: Compare compound and simple interest.
MMOD.4.7: Identify the formula to compute compound interest.
MMOD.4.8: Identify the formula to compute simple interest.

Prior Knowledge Skills:
• Evaluate a function rule given the independent variable.
• Define arithmetic sequence, geometric sequence, and input-output pairs.
• Define sequences and recursively-defined sequences.
• Recognize that sequences are functions whose domain is the set of all positive integers and zero
• Calculate the common ratio of a geometric sequence.
 Mathematics (2019) Grade(s): 9 - 12 Mathematical Modeling All Resources: 0
5. Compare simple and compound interest, and straight-line and proportional depreciation.

Unpacked Content Evidence Of Student Attainment:
Students:
• Can calculate simple interest on investments or loans.
• Can calculate compound interest on loans or investments.
• Can compare or contrast the advantages of simple interest vs compound interest.
• Can compare or contrast the advantages of straight line depreciation vs. proportional depreciation.
Teacher Vocabulary:
• Simple Interest
• Compound Interest
• Straight Line Depreciation
• Proportional Depreciation
Knowledge:
Students know:
• how to calculate both simple interest and compound interest and straight line depreciation and proportional depreciation.
Skills:
Students are able to:
• Create tables that compare interest paid/owed on accounts using simple interest and compound interest.
• Create tables that compare depreciation on items using straight line depreciation and proportional depreciation.
Understanding:
Students understand that:
• interest can be calculated in different ways and there are advantages and disadvantages to each method. (earning interest vs. paying interest).
• Depreciation can be calculated in different ways and there are advantages and disadvantages to each method.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
MMOD.5.1: Define simple and compound interest, and straight-line and proportional depreciation.
MMOD.5.2: Compare simple and compound interests.
MMOD.5.3: Identify the formula to compute compound interest.
MMOD.5.4: Identify the formula to compute simple interest.
MMOD.5.5: Compare straight-line and proportional depreciation.

Prior Knowledge Skills:
• M.7.3.1: Define interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, and percent error.
• M.7.3.2: Apply definitions to context in real world problems.
 Mathematics (2019) Grade(s): 9 - 12 Mathematical Modeling All Resources: 0
6. Investigate growth and reduction of credit card debt using spreadsheets, including variables such as beginning balance, payment structures, credits, interest rates, new purchases, finance charges, and fees.

Unpacked Content Evidence Of Student Attainment:
Students:
• Can use interest rate, credits, new purchases, finance charges and fees to calculate the balance on a credit card over time.
• Can set up a spreadsheet to calculate a credit card balance.
Teacher Vocabulary:
• Beginning Balance
• Payment Structure
• Investment Rate
• Finance Charge
• Fee
Knowledge:
Students know:
• how to calculate the balance on a credit card.
Skills:
Students are able to:
• Calculate the growth of credit card debt or the reduction of credit card debt.
• Recognize that payments or new purchases affect the balance on a credit card.
• Recognize that annual fees and interest rates affect the balance on a credit card.
• Recognize that making a minimum payment does not significantly reduce the balance on a credit card.
• Use a spreadsheet to perform repetitive calculations.
Understanding:
Students understand that:
• Credit card balances can grow due to high interest rates, finance charges, fees and new purchases.
• Care should be taken to insure that credit card balances are monitored and controlled.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
MMOD.6.1: Define previous balance, payments, credits, interest rate, finance charge fees, credit score, exponential growth, and exponential decay.
MMOD.6.2: Use exponential growth and exponential decay to model given relationships between quantities.
MMOD.6.3: Calculate cost of credit card interest with benefits.
MMOD.6.4: Discuss cause and effect between use of credit and personal credit score.
MMOD.6.5: Calculate a finance charge at various percentages.
MMOD.6.6: Assess a monthly credit card statement.
MMOD.6.7: Identify benefits associated with credit cards.
MMOD.6.8: Identify the long-term costs of borrowing money.

Prior Knowledge Skills:
• Experience with checking and savings accounts.
• Real-world examples of credit cards
• Recall the formula of an exponential function.
• Recall the slope-intercept form of a linear function.
• Define b as growth or decay factor in the context of an exponential problem.
• Define k as the initial amount in the context of an exponential problem.
 Mathematics (2019) Grade(s): 9 - 12 Mathematical Modeling All Resources: 0
7. Compare and contrast housing finance options including renting, leasing to purchase, purchasing with a mortgage, and purchasing with cash.

a. Research and evaluate various mortgage products available to consumers.

b. Compare monthly mortgage payments for different terms, interest rates, and down payments.

c. Analyze the financial consequence of buying a home (mortgage payments vs. potentially increasing resale value) versus investing the money saved when renting, assuming that renting is the less expensive option.

Unpacked Content Evidence Of Student Attainment:
Students:
• Can calculate the cost of housing through rent, rent to own, mortgage or cash payments.
• Can compare these costs and make a decision about the best option in a given situation.
Teacher Vocabulary:
• Rent
• Lease to Purchase
• Mortgage
• Interest Rate
• Down Payment
Knowledge:
Students know:
• how to calculate the cost of renting a house for a period of time.
• how to calculate the cost of purchasing a house with a mortgage.
• how to organize data on housing costs.
• how to compare and contrast housing cost data.
Skills:
Students are able to:
• Calculate the cost of renting a house.
• Calculate the cost of renting to own a house.
• Calculate the cost of a house through a mortgage purchase.
• Calculate the cost of a house through a cash purchase.
• Compare and contrast the cost of purchasing a home using different methods.
Understanding:
Students understand that:
• there are many ways to pay for housing. Each method has advantages and disadvantages for individuals.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
MMOD.7.1: Define mortgage and lease.
MMOD.7.2: Identify various types of mortgages.
Examples: 30-year loan, 15-year loan, fixed rate loans, adjustable rate loans, VA loans, FHA loans.
MMOD.7.3: Investigate housing costs in local area.
MMOD.7.4: Identify housing options.

Prior Knowledge Skills:
• Identify different types of housing in local community.
• Use of internet to find local house prices
 Mathematics (2019) Grade(s): 9 - 12 Mathematical Modeling All Resources: 0
8. Investigate the advantages and disadvantages of various means of paying for an automobile, including leasing, purchasing by cash, and purchasing by loan.

Unpacked Content Evidence Of Student Attainment:
Students:
• Can calculate the cost of an automobile using a lease plan.
• Can calculate the cost of an automobile using a cash purchase.
• Can calculate the cost of an automobile using a loan/payment plan.
• Can compare these costs and make an informed decision about the best option in a given situation.
Teacher Vocabulary:
• Lease
• Principal
• Interest Rate
• Compounding Periods
Knowledge:
Students know:
• how to calculate the cost of leasing a car.
• how to calculate the cost of an automobile loan.
• how to organize data on car purchases.
• how to compare and contrast the car purchase data.
Skills:
Students are able to:
• Use the compound interest formula to calculate the cost of purchasing a car through a loan.
• Calculate the cost of a car when leased.
• organize car purchase data.
• Compare and contrast data to make an informed decision.
Understanding:
Students understand that:
• there are different methods for purchasing a car. Each method has advantages and disadvantages for individuals.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
MMOD.8.1: Define depreciation and leasing.
MMOD.8.2: Compare the cost of purchasing a vehicle by cash, leasing, and by loan.
MMOD.8.3: Determine the cost of purchasing a vehicle with cash.
MMOD.8.4: Determine the cost of purchasing a vehicle by leasing.
MMOD.8.5: Determine the cost of purchasing a vehicle by loan.

Prior Knowledge Skills:
• Use of interent to determine price of vehicles at local dealerships.
• Knowledge of payment types through commercials.
Two- and three-dimensional representations, coordinates systems, geometric transformations, and scale models are useful tools in planning, designing, and constructing solutions to real-world problems.
 Mathematics (2019) Grade(s): 9 - 12 Mathematical Modeling All Resources: 0
9. Use the Mathematical Modeling Cycle to solve real-world problems involving the design of three-dimensional objects.

Unpacked Content Evidence Of Student Attainment:
Students:
• Can use surface area and volume formulas for three-dimensional figures.
• Can create a mathematical model and use it to solve a design problem.
Teacher Vocabulary:
• Mathematical Modeling Cycle
• Three Dimensional Object
Knowledge:
Students know:
• the surface area formulas for cylinders, pyramids, cones and spheres.
Skills:
Students are able to:
• Calculate the surface area for cylinders, pyramids, cones and spheres.
• Calculate volume for cylinders, pyramids, cones and spheres.
• Use the mathematical modeling cycle
Understanding:
Students understand that:
• Surface area and volume can be used to approximate or solve real-world problems involving three dimensional figures.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
MMOD.9.1: Define three-dimensional, scale factor, and transformations.
MMOD.9.2: Define the problem to be answered.
MMOD.9.3: Make assumptions to simplify the situation.
MMOD.9.4: Identify variables in the situation, and select those that represent essential features in order to formulate a mathematical model.
MMOD.9.5: Analyze and perform operations to draw conclusions.
MMOD.9.6: Assess the model and solutions in terms of the original situation.
MMOD.9.7: Refine and extend the model as needed.
MMOD.9.8: Report on the conclusions and the reasoning.

Prior Knowledge Skills:
• Compare and contrast the random sampling data to the population.
• Analyze conclusions of the sample to determine its appropriateness for the population.
• Predict an outcome of the entire population based on random samplings.
• Justify the mathematical and statistical reasoning.
 Mathematics (2019) Grade(s): 9 - 12 Mathematical Modeling All Resources: 0
10. Construct a two-dimensional visual representation of a three-dimensional object or structure.

a. Determine the level of precision and the appropriate tools for taking the measurements in constructing a two-dimensional visual representation of a three-dimensional object or structure.

b. Create an elevation drawing to represent a given solid structure, using technology where appropriate.

c. Determine which measurements cannot be taken directly and must be calculated based on other measurements when constructing a two-dimensional visual representation of a three-dimensional object or structure.

d. Determine an appropriate means to visually represent an object or structure, such as drawings on paper or graphics on computer screens.
Unpacked Content Evidence Of Student Attainment:
Students:
• Can identify a three dimensional geometric figure that can be used to describe an object.
• Can connect a three dimensional object to a net comprised of two dimensional shapes.
• Can create an elevation drawing to represent a solid structure.
Teacher Vocabulary:
• Two Dimensional
• Three Dimensional
• Precision
• Elevation Drawings
Knowledge:
Students know:
• how to create a net comprised of two dimensional objects for a three dimensional figure.
• how to describe a three dimensional object using a two dimensional cross-section or a rotation of a two dimensional object.
Skills:
Students are able to:
• Recognize two dimensional shapes and use those to create nets of three dimensional objects.
• Find crucial measurements of three dimensional objects such as the height of the object, the length, the width or the radius of the base.
Understanding:
Students understand that:
• A three dimensional object is comprised of two dimensional figures.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
MMOD.10.1: Define two-dimensional figure, three-dimensional figure, precision, area, elevation drawing, scale factor, model, and perimeter.
MMOD.10.2: Calculate precise measurements.
MMOD.10.3: Describe the relationship between two- and three-dimensional figures.
MMOD.10.4: Identify appropriate tools for taking measurements of various objects.

Prior Knowledge Skills:
• Define two-dimensional figure, three-dimensional figure, and plane section.
• List attributes of three-dimensional figures.
• List attributes of two-dimensional figures.
• Describe the relationship between two- and three-dimensional figures.
• Define scale factor, similarity and proportions.
• Compare two figures in terms of similarity.
• Create proportional equations from given information.
• Solve proportional equations.
• Prove that equivalent ratios are proportions.
 Mathematics (2019) Grade(s): 9 - 12 Mathematical Modeling All Resources: 0
11. Plot coordinates on a three-dimensional Cartesian coordinate system and use relationships between coordinates to solve design problems.

a. Describe the features of a three-dimensional Cartesian coordinate system and use them to graph points.

b. Graph a point in space as the vertex of a right prism drawn in the appropriate octant with edges along the x, y, and z axes.

c. Find the distance between two objects in space given the coordinates of each.

Examples: Determine whether two aircraft are flying far enough apart to be safe; find how long a zipline cable would need to be to connect two platforms at different heights on two trees.

d. Find the midpoint between two objects in space given the coordinates of each.

Example: If two asteroids in space are traveling toward each other at the same speed, find where they will collide.
Unpacked Content Evidence Of Student Attainment:
Students:
• Can identify the x, y, and z axis on a three dimensional coordinate system.
• Can plot coordinates on the three dimensional coordinate system.
• Can find the distance between points in space.
• Can find the midpoint between points in space.
Teacher Vocabulary:
• Three Dimensional cartesian coordinate system
• Two dimensional cartesian coordinate system
• Points in Space
• Vertex
• Right Prism
• Octant
Knowledge:
Students know:
• how to plot points in two dimensions.
• how to find the distance between two dimensional points.
• how to find the midpoint between two-dimensional point.
Skills:
Students are able to:
• Extend their knowledge of the two dimensional coordinate system to the three dimensional coordinate system.
Understanding:
Students understand that:
• points in space are a part of the three dimensional coordinate system.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
MMOD.11.1: Define two-dimensional and three-dimensional Cartesian coordinate systems.
MMOD.11.2: Determine how to graph a point in a three-dimensional coordinate system.
MMOD.11.3: Calculate the distance between two objects in space.
MMOD.11.4: Calculate the midpoint between two objects in space.
MMOD.11.5: Compare and contrast a three-dimensional and two-dimensional Cartesian coordinate system.
MMOD.11.6: Determine how to graph a point in a two-dimensional coordinate system.
MMOD.11.7: Calculate the distance between two objects.
MMOD.11.8: Calculate the midpoint between two objects.
MMOD.11.9: Identify a diagram that shows a two-dimensional and three-dimensional

Prior Knowledge Skills:
• Identify ordered pairs.
• Recognize ordered pairs.
• Define ordered pair and coordinate plane.
• Create linear equations with two variables.
• Graph linear equations on coordinate axes with labels and scales.
• Identify an ordered pair and plot it on the coordinate plane.
 Mathematics (2019) Grade(s): 9 - 12 Mathematical Modeling All Resources: 0
12. Use technology and other tools to explore the results of simple transformations using three-dimensional coordinates, including translations in the x, y, and/or z directions; rotations of 90°, 180°, or 270° about the x, y, and z axes; reflections over the xy, yz, and xy planes; and dilations from the origin.

Example: Given the coordinates of the corners of a room in a house, find the coordinates of the same room facing a different direction.
Unpacked Content Evidence Of Student Attainment:
Students:
• Can preform simple transformations on three dimensional coordinates
Teacher Vocabulary:
• Transformation
• Translation
• Rotation
• Three Dimensional Coordinates
Knowledge:
Students know:
• how to translate points in the two dimensional coordinate plane and can extend this knowledge to three dimensions.
• how to rotate points 90, 180 or 270 degrees in the two dimensional coordinate plane and can extend this knowledge to the three dimensional coordinate system.
• how to reflect two dimensional objects over the x- or y-axis and can extend this knowledge to the three dimensional coordinate system.
Skills:
Students are able to:
• extend their knowledge of transformations in the two dimensional coordinate plane to the three dimensional coordinate system.
Understanding:
Students understand that: transformations can be performed in the three dimensional coordinate systems and the transformations are similar to those in the two dimensional coordinate plane.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
MMOD.12.1: Define translation, rotation, and dilation.
MMOD.12.2: Use technology or other tools to interpret the results of transformations (translation, rotation, and dilation).
MMOD.12.3: Use technology or other tools to rotate an object 90o or 180o.
MMOD.12.4: Use technology or other tools to dilate an object.
MMOD.12.5: Use technology or other tools to translate an object.
MMOD.12.6: Use technology or other tools to reflect an object.

Prior Knowledge Skills:
• Define dilation.
• Recall how to find scale factor.
• Give examples of scale drawings.
• Recognize translations.
• Recognize reflections.
• Recognize rotations.
 Mathematics (2019) Grade(s): 9 - 12 Mathematical Modeling All Resources: 0
13. Create a scale model of a complex three-dimensional structure based on observed measurements and indirect measurements, using translations, reflections, rotations, and dilations of its components.

Example: Develop a plan for a bridge structure using geometric properties of its parts to determine unknown measures and represent the plan in three dimensions.
Unpacked Content Evidence Of Student Attainment:
Students:
• Can find measurements both observed and indirect from two dimensional objects.
• Can combine two dimensional objects to form a three dimensional figure.
• Can use dilations of two dimensional components to produce a scale model of a three dimensional object.
Teacher Vocabulary:
• Two Dimensional Object
• Three Dimensional Object
• Translation
• Reflection
• Rotation
• Dilation
Knowledge:
Students know:
• that three dimensional figures are formed from two dimensional figures.
• That a dilation produces a smaller or larger version of a figure.
Skills:
Students are able to:
• Combine three dimensional figures to form a scale model.
• Transform a two or three dimensional figure.
• Find a dilation of a three dimensional figure.
Understanding:
Students understand that:
• A scale model of a three dimensional figure is comprised of two dimensional objects that have been transformed.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
MMOD.13.1: Define observed measurements, indirect measurements, scale models, complex three-dimensional shapes, translations, reflections, rotations, and dilations.
MMOD.13.2: Apply geometric concepts in modeling situations.
MMOD.13.3: Perform all transformations (i.e., translations, reflections, rotations, dilations).
MMOD.13.4: Calculate scale factor.

Prior Knowledge Skills:
>
• Define dilation.
• Recall how to find scale factor.
• Give examples of scale drawings.
• Recognize translations.
• Recognize reflections.
• Recognize rotations.
Creating Functions to Model Change in the Environment and Society
Functions can be used to represent general trends in conditions that change over time and to predict future conditions based on present observations.
 Mathematics (2019) Grade(s): 9 - 12 Mathematical Modeling All Resources: 0
14. Use elements of the Mathematical Modeling Cycle to make predictions based on measurements that change over time, including motion, growth, decay, and cycling.

Unpacked Content Evidence Of Student Attainment:
Students:
• Can use motion models to make predictions.
• Can use growth models to make predictions.
• Can use decay models to make predictions.
• Can use cycling models to make predictions.
Teacher Vocabulary:
• Mathematical Modeling Cycle
• Uniform Motion
• Growth Model
• Decay Model
• Cycling
Knowledge:
Students know:
• how to measure motion as it changes over time.
• how to use an exponential function to model growth and decay.
• how to use a Sine or Cosine Function to model cycling functions.
Skills:
Students are able to:
• Calculate motion at any time (d=rt).
• Calculate exponential growth.
• Calculate exponential decay.
• Calculate sine or cosine values from a function.
Understanding:
Students understand that:
• Some problems involve rates that change over time and mathematical models can be used to make predictions.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
MMOD.14.1: Define motion, growth, decay, and cycling.
MMOD.14.2: Define the problem to be answered.
MMOD.14.3: Make assumptions to simplify the situation.
MMOD.14.4: Identify variables in the situation, and select. those that represent essential features in order to formulate a mathematical model.
MMOD.14.5: Analyze and performing operations to draw conclusions.
MMOD.14.6: Assess the model and solutions in terms of the original situation.
MMOD.14.7: Refine and extend the model as needed.
MMOD.14.8: Report on the conclusions and the reasoning.

Prior Knowledge Skills:
• Solve the equation represented by the real-world situation.
• Set up an equation to represent the given situation, using correct mathematical operations and variables.
• Given a contextual situation, interpret and defend the solution in the context of the original problem.
• Define equation, expression, variable, equality and inequality.
 Mathematics (2019) Grade(s): 9 - 12 Mathematical Modeling All Resources: 0
15. Use regression with statistical graphing technology to determine an equation that best fits a set of bivariate data, including nonlinear patterns.

Examples: global temperatures, stock market values, hours of daylight, animal population, carbon dating measurements, online streaming viewership

a. Create a scatter plot with a sufficient number of data points to predict a pattern.

b. Describe the overall relationship between two quantitative variables (increase, decrease, linearity, concavity, extrema, inflection) or pattern of change.

c. Make a prediction based upon patterns.
Unpacked Content Evidence Of Student Attainment:
Students:
• Can use graphing technology to plot a scatter diagram.
• Can describe a relationship if one exists between points.
• Can find a regression equation using graphing technology.
• Can use the regression equation to make a prediction.
Teacher Vocabulary:
• Regression Equation
• "Best Fit"
• Bivariate Data
• Linear Pattern
• Non-linear pattern
• Scatter Plot
• Quantitative Variable
• Extrema
• Inflection
Knowledge:
Students know:
• how to plot points using graphing technology.
• how to find a regression equation using graphing technology.
• how to use a regression equation to make a prediction.
Skills:
Students are able to:
• plot points.
• Distinguish between linear and nonlinear functions.
• Use graphing technology.
Understanding:
Students understand that:
• Regression equations can be used to model data.
• Graphing technology helps us find regression equations.
• The regression equation can be used to make a prediction.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
MMOD.15.1: Define bivariate scatter plot, outlier, cluster, linear, nonlinear, positive and negative association, slope, intercept, linear, equation, concave up, concave down, and bivariate.
MMOD.15.2: Make a prediction based upon patterns.
MMOD.15.3: Describe patterns found in a scatter plot.
MMOD.15.4: Demonstrate how to label and plot information on a scatter plot (dot plot).
MMOD.15.5: Distinguish the difference between positive and negative correlation.
MMOD.15.6: Use technology to find the equation of a line given data points.

Prior Knowledge Skills:
• Define bivariate scatter plot, quantitative data, outlier, cluster, linear, nonlinear, and positive and negative association.
• Describe patterns found in a scatter plot.
• Demonstrate how to label and plot information on a scatter plot (dot plot).
• Distinguish the difference between positive and negative correlation.
• Recall how to describe the spread of the scatter plot (dot plot).
• Create a scatter plot and line of best fit using data from a spreadsheet.
• Organize and display bivariate quatitative data using a scatter plot, and extend from simple cases by hand to more complex cases involving a large data set using technology.
• Create a scatter plot of data.
• Calculate the fit of the function to the data by examining residuals.
 Mathematics (2019) Grade(s): 9 - 12 Mathematical Modeling All Resources: 0
16. Create a linear representation of non-linear data and interpret solutions, using technology and the process of linearization with logarithms.

Unpacked Content Evidence Of Student Attainment:
Students:
• Can use logarithmic properties to create a linear representation of non-linear data.
Teacher Vocabulary:
• Non-linear
• Linearization
Knowledge:
Students know:
• the properties of logarithms and can use them to rewrite problems in different forms.
Skills:
Students are able to:
• use the power property of logarithms.
• Recognize the difference between a linear and a nonlinear function.
Understanding:
Students understand:
• It is often easier to solve linear functions rather than non-linear functions.
• The use of logarithmic properties allows you to rewrite exponential functions as linear functions.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
MMOD.16.1: Define linearization, linear, non-linear, exponential function and logarithmic function.
MMOD.16.2: Interpret solutions based on results.
MMOD.16.3: Using technology create a linear representation of nonlinear data.
MMOD.16.4: Using technology graph a logarithmic function.
MMOD.16.5: Using technology graph an exponential function.

Prior Knowledge Skills:
• Define logarithmic and exponential function.
• Recognize the inverse relationship of logarithmic function and exponential functions.
• Define ordered pair, coordinate plane, polynomial, trigonometric (sine and cosine), logarithmic, reciprocal, radical, and general piecewise functions.
• Create equations with two variables (polynomial, trigonometric (sine and cosine), logarithmic, reciprocal, radical, and general piecewise functions).
• Graph equations on coordinate axes with labels and scales (polynomial, trigonometric (sine and cosine), logarithmic, reciprocal, radical, and general piecewise functions.).
• Define linear function and exponential function.
• Distinguish between graphs of a line and an exponential function.
• Identify the graph of an exponential function.
Modeling to Interpret Statistical Studies
Statistical studies allow a conclusion to be drawn about a population that is too large to survey completely or about cause and effect in an experiment.
 Mathematics (2019) Grade(s): 9 - 12 Mathematical Modeling All Resources: 0
17. Use the Statistical Problem Solving Cycle to answer real-world questions.

Unpacked Content Evidence Of Student Attainment:
Students:
• Can define a problem and create an investigative question.
• Can create a plan for what variables to examine and how they should be examined.
• Can collect and organize data.
• Can analyze the collected data.
• Can reach a conclusion about the problem.
Teacher Vocabulary:
• Statistical Problem Solving Cycle
Knowledge:
Students know:
• that statistical problems can be solved using a systematic approach.
Skills:
Students are able to:
• Collect and organize data.
• Use graphs and charts to summarize data.
• Communicate their finding to others.
Understanding:
Students understand that:
• Statistical problems can be solved using the statistical problem solving cycle.
• Real-world problems can be solved using data that has been collected, organized and analyzed.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
MMOD.17.1: Formulate question.
MMOD.17.2: Design study.
MMOD.17.3: Collect data.
MMOD.17.4: Communicate interpretations and limitations.
MMOD.17.5: Interpret, refine variables and assumptions.
MMOD.17.6: Analyze results.

Prior Knowledge Skills:
• Solve the equation represented by the real-world situation.
• Set up an equation to represent the given situation, using correct mathematical operations and variables.
• Given a contextual situation, interpret and defend the solution in the context of the original problem.
• Define equation, expression, variable, equality and inequality.
 Mathematics (2019) Grade(s): 9 - 12 Mathematical Modeling All Resources: 0
18. Construct a probability distribution based on empirical observations of a variable.

Example: Record the number of student absences in class each day and find the probability that each number of students will be absent on any future day.

a. Estimate the probability of each value for a random variable based on empirical observations or simulations, using technology.

b. Represent a probability distribution by a relative frequency histogram and/or a cumulative relative frequency graph.

c. Find the mean, standard deviation, median, and interquartile range of a probability distribution and make long-term predictions about future possibilities. Determine which measures are most appropriate based upon the shape of the distribution.
Unpacked Content Evidence Of Student Attainment:
Students:
• Can collect information or data from something they have experienced and create a probability distribution.
• Can calculate the mean, standard deviation, median and interquartile range of a set of data.
• Can make long term predictions based on the calculations and the data.
Teacher Vocabulary:
• Probability Distribution
• Empirical Observation
• Random Variable
• Relative Frequency Histogram
• Cumulative Relative Frequency Graph
• Mean
• Median
• Standard Deviation
• Interquartile Range
Knowledge:
Students know:
• how to collect data from an experiment or experience.
• how to create a probability distribution finding the probability of each event.
• how to create a graph or a table to display the data.
Skills:
Students are able to:
• Find statistical measures mean, median, standard deviation and interquartile range.
• Create histograms.
• Collect and organize observed data.
Understanding:
Students understand that:
• A probability distribution can be used to display statistical data. -Statistical calculations and data can be used to make long-term predictions.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
MMOD.18.1: Define center, mean, median, spread, interquartile range, standard deviation, data set, dot plots, histograms, empirical observations and box plots.
MMOD.18.2: Make long-term predictions based on the calculations.
MMOD.18.3: Find the mean, standard deviation, median, and interquartile range. Determine which measures are most appropriate based upon the shape of the distribution.
MMOD.18.4: Represent the probability distribution by a relative frequency histogram and/or a cumulative relative frequency graph.
MMOD.18.5: Find the probability of each value for the random variable.

Prior Knowledge Skills:
• Define normal distribution, mean, standard deviation, and empirical rule.
• Use technology to calculate mean and standard deviation.
• Use technology (ex. calculator, Microsoft Excel, etc.) to estimate areas under the normal curve.
• Analyze data sets to determine if appropriate.
• Accurately find the center (median and mean) and spread (interquartile range and standard deviation) of data sets,
• Present viable arguments and critique arguments of others from the comparison of the center and spread of multiple data sets.
• Reason how standard deviation develops from the mean absolute deviation.
• Define probability, ratio, simple event, compound event, and independent event.
• Determine the probability of a compound event.
• Determine the probability of an independent event.
• Determine the probability of a simple event by expressing the probability as a ratio, percent, or decimal.
• Identify the probability of an event that is certain as 1 or impossible as 0.
• Solve word problems involving probability.
• Use proportional relationships to solve mulit-step ratio and percent problems.
• Recognize and represent proportional relationships as ratios between two quantities.
 Mathematics (2019) Grade(s): 9 - 12 Mathematical Modeling All Resources: 0
19. Construct a sampling distribution for a random event or random sample.

Examples: How many times do we expect a fair coin to come up "heads" in 100 flips, and on average how far away from this expected value do we expect to be on a specific set of flips? What do we expect to be the average height for a random sample of students in a local high school given the mean and standard deviation of the heights of all students in the high school?

a. Use the binomial theorem to construct the sampling distribution for the number of successes in a binary event or the number of positive responses to a yes/no question in a random sample.
b. Use the normal approximation of a proportion from a random event or sample when conditions are met.

c. Use the central limit theorem to construct a normal sampling distribution for the sample mean when conditions are met.

d. Find the long-term probability of a given range of outcomes from a random event or random sample.
Unpacked Content Evidence Of Student Attainment:
Students:
• Can construct a graph based on one statistic from an event.
• Can use the graph to find the long term probability from a range of outcomes.
Teacher Vocabulary:
• Sampling Distribution
• Random Event
• Random Sample
• Binomial Theorem
• Binary Event
• Normal Approximation
• Central Limit Theorem
• Normal Sampling Distribution
• Sample Mean
• Binomial Distribution
Knowledge:
Students know:
• how to use the binomial theorem.
• how to setup a normal distribution.
Skills:
Students are able to:
• Find the mean and standard deviation for a set of data.
• Create a normal distribution for a set of data.
• Read and interpret the normal curve.
Understanding:
Students understand that:
• Collected data can be organized in a way to allow interpretation and prediction. Some data is considered normally distributed while other data sets are skewed.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
MMOD.19.1: Define sample, validity, population, inference, random sampling, statistic, binomial theorem, binary event, generalization, normal approximation of a proportion, central limit theorem and normal sampling distribution.
MMOD.19.2: Explain the validity of random sampling.
MMOD.19.3: Differentiate the appropriate sampling method.
MMOD.19.4: Analyze attributes of sample size.
MMOD.19.5: Draw conclusions by finding the long-term probability of a given range of outcomes from a random event or random sample.

Prior Knowledge Skills:
• Define mean, standard deviation, population, sample, and correlation coefficient.
• Define sample, validity, population, inference, random sampling, statistic, and generalization.
• Identify the nature of the attribute, how it was measured, and its unit of measure.
• Discuss real world examples of valid sampling and generalizations.
• Compare sample size with population to check for validity.
• Analyze attributes of sample size.
• Differentiate between appropriate sampling methods.
• Explain the validity of random sampling.
 Mathematics (2019) Grade(s): 9 - 12 Mathematical Modeling All Resources: 0
20. Perform inference procedures based on the results of samples and experiments.

a. Use a point estimator and margin of error to construct a confidence interval for a proportion or mean.

b. Interpret a confidence interval in context and use it to make strategic decisions.

Example: short-term and long-term budget projections for a business

c. Perform a significance test for null and alternative hypotheses.

d. Interpret the significance level of a test in the context of error probabilities, and use the results to make strategic decisions.

Example: How do you reduce the rate of human error on the floor of a manufacturing plant?
Unpacked Content Evidence Of Student Attainment:
Students:
• Can construct a confidence interval for the results of a statistical sample or experiment.
• Can use the confidence interval to make a decision.
• Can perform a significance test for null and alternative hypothesis.
• Can interpret a significance test and us the result to make a decision.
Teacher Vocabulary:
• Point Estimator
• Margin of Error
• Confidence Interval
• Significance Test
• Null Hypothesis
• Hypothesis
Knowledge:
Students know:
• how to calculate the margin of error in a statistical sample.
• how to express the confidence interval for a statistical sample or experiment.
• how to perform a significance test.
• how to use the results of the significance test to either support or refute a claim.
Skills:
Students are able to:
• Calculate the margin of error.
• Determine a confidence interval.
• perform a significance test.
• Use the results of the significance test to support or refute the null hypothesis.
Understanding:
Students understand that:
• hypothesis testing is used to evaluate claims about a population.
• a confidence interval helps to determine the size of a sample needed to provide accurate calculations.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
MMOD.20.1: Define samples, inference, experiments, point estimator, margin of error, confidence interval, proportion, mean, null and alternative hypotheses, significance test and error of probabilities.
MMOD.20.2: Interpret the significance level of a test in the context of given error probabilities.
MMOD.20.3: Differentiate the appropriate sampling method.
MMOD.20.4: Given a point estimator and margin of error, determine confidence interval.
MMOD.20.5: Given data, perform and interpret a significance test for null alternative hypotheses.
MMOD.20.6: Use the given results to make strategic decisions.
MMOD.20.7: Collect and organize data for analysis.

Prior Knowledge Skills:
• Identify the attribute used to create the numerical set.
• Organize the data.
• Collect the data.
• Compare and contrast the center and variation.
• Define numerical data set, quantitative, measure of center, median, frequency distribution, and attribute.
• Define margin of error and confidence interval.
• Justify the mathematical and statistical reasoning.
 Mathematics (2019) Grade(s): 9 - 12 Mathematical Modeling All Resources: 0
21. Critique the validity of reported conclusions from statistical studies in terms of bias and random error probabilities.

Unpacked Content Evidence Of Student Attainment:
Students:
• Can construct valid arguments to either defend or refute conclusions from statistical studies.
Teacher Vocabulary:
• Validity
• Bias
• Random Error Probability
Knowledge:
Students know:
• what constitutes a bias in a statistical study.
• The accepted statistical process that can be used to analyze results from a statistical study.
Skills:
Students are able to:
• Calculate and interpret results from a statistical study.
• Calculate random error probability.
• Identify biases that can affect the validity of a mathematical argument.
Understanding:
Students understand that:
• A valid mathematical argument is based on rigorous statistical processes.
• bias and random error probability can affect the validity of a mathematical argument.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
MMOD.21.1: Define validity, conclusions, bias and random error probabilities.
MMOD.21.2: Critique the validity of reported conclusions.
MMOD.21.3: Describe processes that can be used to make fair decisions.

Prior Knowledge Skills:
• Define and discuss bias.
• Compare and contrast statistical situations to determine if statistical bias exists.
• Define bias (sampling, response, or nonresponse bias).
• Interpret survey results.
• Determine where bias may occur.
 Mathematics (2019) Grade(s): 9 - 12 Mathematical Modeling All Resources: 0
22. Conduct a randomized study on a topic of student interest (sample or experiment) and draw conclusions based upon the results.

Example: Record the heights of thirty randomly selected students at your high school. Construct a confidence interval to estimate the true average height of students at your high school. Question whether or not this data provides significant evidence that your school's average height is higher than the known national average, and discuss error probabilities.
Unpacked Content Evidence Of Student Attainment:
Students:
• Can define a problem or a question.
• Can identify variables of interest.
• Can devise a measuring technique.
• Can collect data.
• Can analyze data and draw conclusions.
• Can communicate results.
Teacher Vocabulary:
• Randomized Study
• Sample
• Experiment
Knowledge:
Students know:
• how to design a statistical study.
• how to collect data.
• how to construct confidence intervals.
• how to use confidence intervals to make decisions.
• how to conduct a hypothesis test.
• how to use the hypothesis test to make a decision.
• how to communicate the results of their study.
Skills:
Students are able to:
• Collect data appropriately.
• Calculate and interpret results from a statistical study.
• Calculate margin of error.
• Construct confidence intervals.
• Conduct hypothesis tests.
Understanding:
Students understand:
• how to design and conduct a statistical study and are able to communicate their findings to others.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
MMOD.22.1: Define sample, experiment, randomized study, outliers, and scatterplot.
MMOD.22.2: Predict probabilities based on the effect of outliers on the data.
MMOD.22.3: Evaluate and draw conclusions based on the collected data.
MMOD.22.4: Create a model of a set of data. (i.e., Google form, table, curve, scatterplot)

Prior Knowledge Skills:
• Identify outliers for the mean and standard deviation.
• Compare and contrast the random sampling data to the population.
• Analyze conclusions of the sample to determine its appropriateness for the population.
• Predict an outcome of the entire population based on random samplings.
• Justify the mathematical and statistical reasoning.